Quantum gates and quantum algorithms with Clifford algebra technique

TL;DR

This paper introduces a Clifford algebra-based technique for representing quantum gates and algorithms, including a method that reproduces Grover's algorithm, using spinor representations of SO(1,3).

Contribution

The paper presents a novel Clifford algebra approach to model quantum gates and algorithms, providing an elegant mathematical framework for quantum computation.

Findings

Reproduces Grover's algorithm with specific initial states

Uses spinor representations of SO(1,3) for n-qubits

Provides a new algebraic method for quantum information extraction

Abstract

We use our Clifford algebra technique, that is nilpotents and projectors which are binomials of the Clifford algebra objects $\gamma^a$ with the property $\{\gamma^a,\gamma^b\}_+ = 2 \eta^{ab}$, for representing quantum gates and quantum algorithms needed in quantum computers in an elegant way. We identify $n$-qubits with spinor representations of the group SO(1,3) for a system of $n$ spinors. Representations are expressed in terms of products of projectors and nilpotents. An algorithm for extracting a particular information out of a general superposition of $2^n$ qubit states is presented. It reproduces for a particular choice of the initial state the Grover's algorithm.